## La tour de fibrations exactes des n-catégories.(French)Zbl 0562.18007

The author studies some aspects of n-categories. First he produces a basic construction on a full subcategory V’ of a left exact category V, reflective by a left exact functor K. The construction produces a full subcategory $$Cat_ K(V)$$ of the category Cat(V) of internal categories in V; the objects of $$Cat_ K(V)$$ are those internal categories trivialized by K. It turns out that $$Cat_ K(V)$$ is again a left exact category and V is a full subcategory of $$Cat_ K(V)$$, reflective by a left exact functor $$K_ 1$$. Thus the construction can be iterated.
The object of the paper is to prove two results. First, the category of n-categories can be obtained by iterating n times the above construction, starting from $$1\to Sets$$. Second, the functor K involved in the construction is actually a fibration; when it is an exact fibration, so is $$K_ 1$$. As a consequence the author obtains, for every exact category E, a tower of exact fibrations: $\infty -Cat(E)...\to n- Cat(E)\to (n-1)-Cat(E)\to...\to Cat(E)\to E\to 1.$ The author promises to deduce from this, in a further paper, some Dold-Kan theorem for n- categorical abelian groups (equivalence between the category of abelian complexes of length n and the category of internal n-categories in Ab).
Reviewer: F.Borceux

### MSC:

 18G35 Chain complexes (category-theoretic aspects), dg categories 18D30 Fibered categories 18D05 Double categories, $$2$$-categories, bicategories and generalizations (MSC2010) 18E10 Abelian categories, Grothendieck categories
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### References:

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